Fig. 1. A two-phase PH distribution with Coxian representation. Notice that a PH distribution is the distribution of the absorption time in a continuous time Markov chain. The i th state has exponentially-distributed sojourn time with rate μ_i. With probability p_0i we start in the i th state, and the next state is state j with probability p_ij. Each state has some probability of leading to absorption. The absorption time is the sum of the times spent in each of the states.

Fig. 2. Markov chains for cycle stealing, where , K_sw=K_ba=0, and X_B is exponentially distributed with rate μ_B. Part (a) shows a 2D-infinite Markov chain tracking both the number of beneficiary jobs and the number of donor jobs. Part (b) shows a 1D-infinite transition diagram tracking the number of beneficiary jobs and binary information (zero or ≥ 1) on the number of donor jobs, where B_D is represented by a bold transition. Part (c) shows a 1D-infinite Markov chain, where B_D is represented by a two-phase PH distribution with Coxian representation.

Fig. 3. A transition diagram for cycle stealing where , K_sw=K_ba=0, and X_B has a two-phase PH distribution with Coxian representation (Fig. 1). B_D is represented by a bold transition, but this should be replaced by a PH distribution as in Fig. 2(c).

Fig. 4. Transition diagrams used in the analysis of beneficiary mean response time in the general case. Column (a) shows a transition diagram among regions, I₀, I₁₊, B, C_i, and S_i for for the general case of cycle stealing. Column (b) shows a transition diagram for cycle stealing where , , and X_B and K_sw have exponential distributions. Busy periods, B_D and B_2D+ba, are drawn using a single transition, but this should be replaced by PH distributions as in Fig. 2(c).

Fig. 5. (1) E[X_D]=1,E[K]=1; (2) E[X_D]=1,E[K]=10; (3) E[X_D]=10,E[K]=1. Stability condition on beneficiaries for cycle stealing with switching times (K=K_sw=K_ba) and thresholds.

Fig. 6. (a) ρ_B=0.9,ρ_D→0; (b) ρ_B→0,ρ_D=0.6; (c) ρ_B=0.9,ρ_D→1; (d) . Validation of our analysis against four limiting cases, where , , E[K_sw]=E[K_ba]=1, X_B is exponentially-distributed with mean 1, and X_D has a PH distribution with mean 1 and .

Fig. 7. (a) E[X_B]=1,E[X_D]=1; (b) E[X_B]=1,E[X_D]=10; (c) E[X_B]=10,E[X_D]=1. Validation of analysis against simulation. We fix ρ_B=0.9, and vary ρ_L. (Donor response time is not shown for the case of ρ_D=1, since the donor is unstable there.) and . X_B and X_D are exponentially-distributed.

Fig. 8. (a) E[K]=0; (b) E[K]=1; (c) E[K]=0; (d) E[K]=1. The mean response time for beneficiaries and donors as a function of ρ_B under cycle stealing and dedicated servers. In all figures X_B and X_D are exponential with mean 1; switching times are exponential with mean 0 or 1 as labeled.

Fig. 9. (a) E[K]=0; (b) E[K]=0; (c) E[K]=1; (d) E[K]=1. The gain of beneficiary jobs and pain of donor jobs (columns 1 and 3), and the effect of cycle stealing on the overall mean response time relative to dedicated servers (columns 2 and 4). In columns 1 and 3, solid lines delineate high/mid/low gain regions, and dashed lines delineate high/mid/low pain regions. X_B has an exponential distribution; X_D has an exponential distribution in rows 1–3 and a PH distribution with in row 4. The means of X_B and X_D are as labeled.

Fig. 10. The mean response time for beneficiary jobs under different donor job size variability. We fix ρ_D at 0.5, and ρ_B varies from 0 to the stability condition of 1.5. The mean donor job size and beneficiary job size is 1, and switching time is zero.

Fig. 11. (a) E[K]=0; (b) E[K]=1; (c) E[K]=0; (d) E[K]=1. The mean response time for beneficiaries and donors as a function of ρ_B. Graphs show the case of (i) , (ii) and , and (iii) and . In all figures X_B and X_D are exponential with mean 1. Switching times are exponential with mean 0 or 1 as labeled. ρ_D=0.5.

Fig. 12. (a) E[K]=1; (b) E[K]=2; (c) E[K]=1; (d) E[K]=2. Graphs showing the optimal value of and with respect to overall mean response time, where X_B and X_D are exponentially distributed with mean 1.